Factors, Multiples, GCF, and LCM
Factors and multiples are the architecture of the whole numbers. The prime numbers are the atoms: every composite number is built by multiplying primes. Finding the prime factorization of a number exposes its complete structure: 300 = 2^2 x 3 x 5^2. The GCF and LCM, computed from prime factorizations, are the tools needed for fraction arithmetic: to add 3/8 + 5/12, find the LCM of 8 and 12 (= 24) for the common denominator; to simplify 18/24, find the GCF of 18 and 24 (= 6).
Prime factorization
A composite number can be expressed as a product of prime numbers. 60 = 2 x 30 = 2 x 2 x 15 = 2 x 2 x 3 x 5 = 2^2 x 3 x 5. A factor tree systematically breaks composite numbers into prime factors. The result is unique (Fundamental Theorem of Arithmetic): there is only one way to write 60 as a product of primes (up to order).
GCF and LCM from prime factorization
GCF of 36 and 48: 36 = 2^2 x 3^2; 48 = 2^4 x 3. GCF takes the minimum power of each common prime: min(2^2, 2^4) x min(3^2, 3) = 2^2 x 3 = 12. LCM of 36 and 48: take the maximum power: max(2^2, 2^4) x max(3^2, 3) = 2^4 x 3^2 = 144. GCF x LCM = product of the two numbers: 12 x 144 = 1728 = 36 x 48. Verify.
Venn diagrams for common factors and multiples
Venn diagram for factors of 36 and 48: left circle: factors only in 36 (1, 9, 36...); right circle: factors only in 48 (16, 48...); overlap: common factors (1, 2, 3, 4, 6, 12). The largest number in the overlap is the GCF. This visual makes the relationship between the two numbers explicit and memorable.